EE 308 Spring Binary, Hex and Decimal Numbers (4-bit representation) Binary. Hex. Decimal A B C D E F

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1 EE 8 Spring Binary, Hex and Decimal Numbers (-bit representation) Binary Hex 8 9 A B C D E F Decimal 8 9

2 EE 8 Spring What does a number represent? Binary numbers are a code, and represent what the programmer intends for the code. x Some possible codes: r (ASCII) INC (HC instruction).v (Input from A/D converter) (Unsigned number) (Signed number) Set temperature in room to 9 F Set cruise control speed to mph

3 EE 8 Spring Convert Binary to Unsigned Decimal Binary to Unsigned Decimal: x + x + x + x + x + x + x x + x + x + x 8 + x + x + x Hex to Unsigned Decimal Convert Hex to Unsigned Decimal 8D 8 x + x + x + x 8 x 9 + x + x + x 9

4 EE 8 Spring Unsigned Decimal to Hex Convert Unsigned Decimal to Hex Division Q R Decimal Hex / / D / = D

5 EE 8 Spring Unsigned Number Line: Numbers go from to 8 9 Unsigned Number Line Unsigned Number Wheel: Numbers go from to

6 EE 8 Spring Signed Number Line: Numbers go from to Number Wheel: Numbers go from to

7 EE 8 Spring Number Wheel: Carry and Overflow Carry applies to unsigned numbers when adding or subtracting, result is incorrect. Overflow applies to signed numbers when adding or subtracting, result is incorrect. Carry Overflow Blue: Unsigned Numbers Red: Signed Numbers

8 EE 8 Spring Binary, Hex and Decimal (Signed & Unsigned) Numbers (-bit representation) Binary Hex US Decimal S 8 9 A B C D E F

9 EE 8 Spring Signed Number Representation in s Complement Form: If most significant bit is (most significant hex digit ), number is positive. Get decimal equivalent by converting number to decimal, and using + sign. Example for 8 bit number: A > + ( x + x ) + ( x + x ) + 8 If most significant bit is (most significant hex digit 8 F), number is negative. Get decimal equivalent by taking s complement of number, converting to decimal, and using sign. Example for 8 bit number: A > ( D ) ( x + x ) ( x + x ) 9 9

10 EE 8 Spring One s Complement Table Makes It Simple To Find s Complements One s Complement Table F E D C B A 9 8 To take two s complement, add one to one s complement. Take two s complement of DC : FC + = FD

11 EE 8 Spring Overflow and Carry assume you have a fixed word size A carry is generated when you add two unsigned numbers together, and the result is too large to fit in the fixed word size. A carry is generated when you subtract two unsigned numbers, and the result should be negative. An overflow is generated when you add or subtract two signed numbers, and the fixed-length answer has the wrong sign. Addition and Subtraction of Binary and Hexadecimal Numbers ADDITION AND SUBTRACTION OF BINARY AND HEXADECIMAL NUMBERS ) Limit number of digits to specified word size. bit word: + Keep only bits in answer ) Does not matter if numbers are signed or unsigned mechanics the same Do the operation, then determine if carry and/or overflow bits are set. bit word: Neg + Neg Neg Carry is set, overflow is clear

12 EE 8 Spring Condition Code Register Gives Information On Result Of Last Operation S X H I N Z V C Condition Code Register 8 FFs N Negative : most significant bit of result Z Zero V Overflow : > last operation generated an overflow C Carry : > result zero, > result not zero : > last operation generated a carry Note: Not all HC instructions change CCR bits. A bit in the CCR is the result of the last executed instruction which affects that bit. For example, after the instruction sequence: aba ; Add B to A staa $9 ; Store A in address $9 the C bit of the CCR will reflect the result of the ABA instruction, not the result of the STAA instruction.

13 EE 8 Spring Overflow only occurs under certain addition and subtraction operations. S X H I N Z V C Condition Code Register 8 FFs N Negative : most significant bit of result Z Zero : > result zero, > result not zero V Overflow : > last operation generated an overflow C Carry : > last operation generated a carry If you add a positive and a negative number, on overflow never occurs. If you subtract two positive numbers, an overflow never occurs. If you subtract two negative numbers, and overflow never occurs.